DERIVATION OF RECURSION RELATIONS IN SEIBERG WITTEN THEORY

Transcription

1 Università degli Studi di Padova DIPARTIMENTO DI FISICA E ASTRONOMIA Corso di Studi in Fisica Tesi di Laurea Triennale DERIVATION OF RECURSION RELATIONS IN SEIBERG WITTEN THEORY Laureando: Davide Racco Relatore: Prof. Marco Matone Anno Accademico

2

3 Contents Introduction Main features of Seiberg Witten theory 3. Introduction to the path integral formalism Supersymmetry in quantum field theories Action of N = 2 SYM Wilsonian effective action Duality in Seiberg Witten theory U() R symmetry BPS states Riemann surfaces and uniformization theory 5 2. General and topological aspects of Riemann surfaces Covering manifolds Normal forms of Riemann surfaces Differential forms on Riemann surfaces Basis for H (Σ) and period matrix Uniformization theory Uniformization theorem and Liouville equation Relations between the uniformizing group and the fundamental domain Uniformizing equation Algebraic curve associated with the torus Solution of the model and derivation of the recursion relations 3 3. Solution of the model proposed by Seiberg and Witten Connection with algebraic curves and fibration of tori on the moduli space Hypothesis for the expression of a D (u), a(u) Check of the choice by the asymptotic behaviour Derivation of recursion relations Uniformizing equation for the sphere with three punctures Definition of the function G Determination of the recursion relations for the coefficients F k Solution of the model by reflection simmetries of quantum vacua Symmetries of the moduli space Determination of the fundamental domain Bibliography 45 i

4

5 Introduction In 994, the solution achieved by N. Seiberg and E. Witten [] of the effective lagrangian for the supersymmetric field theory with N = 2 and gauge group SU(2) led to remarkable progress in the understanding of non-perturbative properties of supersymmetric fields and string theories. The main aim of the thesis is to expose the solution of the model, together with the mathematical tools necessary for its derivation. In chapter we will introduce, on the basis of the reviews by A. Bilal [2] and W. Lerche [3], the main features of N = 2 supersymmetric SU(2) gauge theory, and the principal steps of the derivation of the Wilsonian effective action after the gauge symmetry breaking because of the Higgs mechanism. Then we will see that the theory admits a remarkable dual description, allowing to investigate its structure in the strong coupling region. Then we will review in chapter 2 many notions and theorems involving complex analysis and the theory of Riemann surfaces [4], since this will be the framework for the solutions of the model through the analysis of the manifold of vacua of the theory, parametrized by the vacuum expectation value of trφ 2, with φ the scalar Higgs. Geometrically, it can be considered as the moduli space M of vacua. Always in this chapter, we will see the link offered by uniformization theory between the study of the uniformizing group for the Riemann surface and differential equations on the surfaces, as presented by Matone [5]. The conclusion of the chapter will show how we can link algebraic curves to Riemann surfaces, in particular the torus, since they will be a crucial point of the solution originally proposed. In chapter 3 we will be able, on the basis of the concepts developed in the previous chapters, to expose the solution of the model proposed by Seiberg and Witten in their original paper []. Then we will show, as noted by Matone [6], how the framework of uniformization theory allows to a different approach for the solution of the model, and to derive the instanton coefficients for the prepotential without their explicit calculation. Both these solutions anyway assume that the moduli space has exactly three singularities, on the basis of strong evidences of the theory. Anyway, as shown in 997 by Bonelli, Matone and Tonin [7], this assumption can be rigorously proved on the basis of two discrete symmetries over the moduli spaces, that follow from two properties that already arised in [6] (where it was assumed that the singularities were three), but that can also be proved from first principles. We will conclude the thesis with the exposition of this proof.

6

7 CHAPTER Main features of Seiberg Witten theory In this chapter we will introduce the basic characteristics of supersymmetric (SUSY) field theories, and we will see how to define a theory with N = 2 supersymmetric charges conserved. For this case, Seiberg and Witten [] in 994 managed to determine the exact form of the effective action, involving the prepotential F, exploiting in particular a duality transformation which can be seen as a generalization of the electromagnetic duality. Before introducing Seiberg Witten theory, we begin by recalling the main idea lying behind the path integral formulation for quantum field theory, introduced by P. A. M. Dirac in 933 and then reviewed and corrected by R. Feynman in Introduction to the path integral formalism The main motivation that led Dirac to investigate a possible alternative interpretation of the action functional in Quantum Mechanics was the particular role that time played in the original formulation by Dirac, von Neumann and Schrödinger. In this formulation, the state of the system at a given initial time t 0 is specified by a state labeled by a ket q t0, and its temporal evolution at time t is given by the ket q t = e i H(t t0) q t0, where H is the hamiltonian operator of the system. This setting clearly shows an asymmetry between spatial and temporal coordinates, which does not exhibit an explicit Lorentz covariance for a relativistic system. Then Dirac began to analyze the role played by the action in classical mechanics, and in particular its peculiarity of being the generator of the canonical transformation that leads the hamiltonian to a vanishing function, giving immediately the equations of the motion of the system. Let us review his euristic deduction [8]. Let us denote with q, p the coordinate and conjugate momentum of a particle, moving for simplicity in one dimension, and with H(q, p) the hamiltonian of the system. We define as canonical transformation a transformation of the conjugate variables, (q, p) (Q, P ), such that the equations of the motion q = H p, ṗ = H q, expressed in the new coordinates on the phase space (Q, P ), can result as the Hamilton s equations for an Hamiltonian H(Q, P ), Q = H P, P = H Q. 3

8 4. Main features of Seiberg Witten theory Given a function S(q, Q, t), if det 2 S q i Q j 0 in its domain of definition, then the equations S(q, Q) P = Q, p = S(q, Q) q, (.) define implicitly a transformation (q, p) (Q, P ) that is canonical (in its definition domain). This can be easily seen since an equivalent condition for a transformation to be canonical is that the Liouville -form pdq is preserved (up to exact forms), and this is the case: P dq pdq = S S dq dq = d( S). Q q Let us see how we can find a generating function that leads to the equation of motion Q = 0, P = 0 (deriving from the hamiltonian H(Q, P ) = 0): in this case the evolution of the system is determined by the equations q = q(q, P, t), p = p(q, P, t), where Q and P are constants that can be determined by the initial conditions of the system. Recalling that the Hamilton s equation are equivalent to the least action principle, and that the Lagrangian function can be expressed as the inverse Legendre transformation of the Hamiltonian, L = p q H, then the variational principle for the system, expressed in the old and new conjugate variables, brings to δ dt(p q H(q, p)) = 0, δ dt(p Q H(Q, P )) = 0. From the previous equations, we can see that the two integrands must differ at most by a total time derivative, p q H(q, p) = P Q H(Q, P ) + df dt ; if we take F as S(q, Q, t), where S(q, Q, t) is the generating function for the canonical transformation, then ds(q, Q, t) = S dt t + S S q + q Q Q = S t + p q P Q, ( = H q, p = S ) q, t = S t. If the last equation is satisfied, then imposing the transformations. the least action principle is automatically fulfilled by the new coordinates (Q, P ) and the hamiltonian H(Q, P ) = 0. This means that we can implicitly write the generating function S by integrating in t the total time derivative of S. Using the previous results and the Hamilton s equation P = 0, we obtain ds dt = S t + S q = H(q, p, t) + p q = L, q S = t t 0 dt L. (.2) Then the action is the generating function of a canonical transformation which transforms the system variables from the initial time t 0 to another time t. We consider now the quantum description of the system with one degree of freedom. The position and momentum operators ˆq, ˆp must satisfy the commutation relations [ˆq, ˆq] = [ˆp, ˆp] = 0, [ˆq, ˆp] = i. (.3) If we assume that at the initial time the particle is in an eigenstate q of the position operator, then ˆq q = q q, and the orthogonality between different eigenstates brings to q q = δ(q q ). In the framework of Quantum Mechanics, we can define a canonical transformation as a transformation

9 .. Introduction to the path integral formalism 5 that does not change the commutation relations.3: then the system will be described in terms of Q states. Now we focus on the matrix elements of the operators ˆq, ˆQ, and their conjugate momenta ˆp = i q, ˆQ = i Q : q ˆq Q = q q Q, q ˆQ Q = Q q Q, q ˆp Q = i q q Q, q ˆP Q = i Q q Q. (.4) We observe that the operators ˆq, ˆQ may not commute, then if we are considering an arbitrary operator F (ˆq, ˆQ) we have to suppose that it can be well-ordered, i. e. it can be written as a sum of terms f (ˆq)f 2 ( ˆQ), such that q f (ˆq)f 2 ( ˆQ) Q = f (q)f 2 (Q) q Q. At this point, Dirac noticed a strong analogy between the quantum mechanical equations.4 and the theory of the canonical transformations. In order to render manifest this analogy, he proposed to set where S is a function of q and Q. It then follows that q Q = e i S(q,Q), (.5) q ˆp Q = S q q Q, q ˆP Q = S q Q. (.6) P Then, assuming that S q between operators: S and Q are well-ordered functions, the equation.6 can be seen as an equality ˆp = S q, ˆP = S Q. Recalling the equations. for the canonical transformation, we see that S, defined by.5, is the quantum equivalent of the generating function. This intuition leads Dirac to apply this analogy in order to suggest a connection between the quantistic transition amplitudes of the system and the action functional: q t0 q t e i t t 0 dt L. (.7) Anyway, this formula is not correct, as we can see using the completeness relations: if we divide the time interval t t 0 into N infinitesimal time intervals t a = t 0 + aε, Nε = t t 0, and denote with q a q ta a complete set of position eigenstates at times t a, then the completeness of the basis of eigenstates brings to q t q t0 = dq dq N q t0 q t q t q t2 q tn q t, while from equation.7, splitting up the integral into N integration regions, we obtain the incorrect formula q t q t0 = q t0 q t q t q t2 q tn q t, which does not include the intermediate integrations. Anyway, if we assume that.7 holds only for infinitesimal time intervals, then we obtain the Feynman Path Integral for the transition amplitudes: { q t q t+δt = A exp i } δtl(q t, q t+δt ), ( N q t0 q t = lim A N N Nε fixed i= = Dq e i S(t0,t,[q]), e i t0 t dtl(q, q) )

10 6. Main features of Seiberg Witten theory where in the last expression the unknown measure Dq must be determined with the canonical quantization of the system. In this formulation, we can see that all possible paths from q t0 to q t contribute to the transition amplitude. Moreover, in the classical limit 0, we see that the oscillating integrand has a significant contribution only from the path that stationarizes the action functional: then the Feynman Path Integral formulation offers a simple way to include the axiom of the least action principle in the theory. Until now, we have seen how the action was introduced in the formulation of quantum mechanics. The main conceptual steps that bring to quantum field theory are essentially the following. First of all, we introduce the lagrangian density, whose integration over the Minkowski space gives the expression of the lagrangian of the theory. Then, the lagrangian is written as a function of fields: this corresponds to the introduction of infinitely many degrees of freedom in the theory. These fields can be distinguished in relation to the corrispondent representation of the Poincaré group they belong to: in this way we can describe scalar (spin 0), spinor (spin 2 ), vector fields (spin ), and so on. Such a formalism allows for the perturbative expansion in Feynman diagrams. Anyway, this is not a topic of this thesis..2 Supersymmetry in quantum field theories In this section we will introduce the framework of SUSY field theories, and show the basic steps that bring to the effective action for N = 2 supersymmetric theory, with gauge group SU(2). The first example of supersymmetric theory was introduced by Wess and Zumino in 974. They considered [8] a lagrangian density that includes only the kinetic terms of two scalar fields S and P and a four component Majorana spinor field χ. Besides the conformal invariance and two independent global phase invariances, they tried to introduce in the lagrangian an invariance for a transformation that changes the spinless fields S and P into the spinor field χ, then mixing bosonic and fermionic degrees of freedom. To fulfill this aim, they introduced, as parameter for the transformation, a Grassmann (anticommuting) spinor variable θ, and two auxiliary scalar fields, in order to ensure that the composition of two such transformations reduces to a separated translation for every field: these auxiliary fields, anyway, can be removed when writing the Euler-Lagrange equations deriving from this lagrangian density. From these ones, moreover, one can see that all the fields (in this case S, P and χ) entering a supermultiplet have the same mass, since they all fulfill the same Klein-Gordon equation. This was the first (and simplest) example of supersymmetry in four dimensions. Since then, many efforts have been made in that direction..2. Action of N = 2 SYM In this section we will introduce basic facts about N = 2 supersymmetric Yang-Mills (SYM) theory [2]. We will consider a theory with symmetry gauge group SU(2). The fields that we will introduce will be considered in the adjoint representation: then, since the generators of the Lie algebra of SU(2) are the j φj σ j. Pauli matrices σ j, when writing a scalar field φ we will mean φ = 2 One of the simplest unextended (N = ) supersymmetries involves a complex scalar field φ and a spinor field ψ α, α =, 2. These fields are assembled in the following superfield Φ: Φ = φ(y) + 2θψ(y) + θ 2 F (y), where θ is the anticommuting spinor variable introduced in the previous section, y µ = x µ + iθσ µ θ (the bar indicates the lowering of the spinor indices θ α = ε αβ θ β ), and F (y) is an auxiliary field. Expanding the y dependence with respect to θ, and performing the integration d 2 θ d 2 θ, one obtains the following action for the first superfield: d 4 x ( µ φ µ φ i ψ σ µ µ ψ + F F ). (.8) The first two terms are standard kinetic terms for a scalar and spinor field, while the auxiliary field F can be set to be equal to zero with Euler-Lagrange equations. Another supersimmetric multiplet is a vector multiplet that contains a vector field A µ, which is the gauge

11 .2. Supersymmetry in quantum field theories 7 field of the theory, with a spinor field superpartner λ α. Using an auxiliary field D, they can be combined together in the spinor superfield W α as W = ( iλ + θd iσ µν θf µν + θ 2 σ µ µ λ)(y), where σ µν = 4 (σµ σ ν σ ν σ µ ), F µν = µ A ν ν A µ ig[a µ, A ν ], µ λ is the covariant derivative µ λ = µ λ ig[a µ, λ] and g is the coupling constant. In this case the supersymmetric action reduces to [ d 4 x d 2 θ W α W α = d 4 x 4 4 F µνf µν + i ] 4 F F µν µν iλσ µ µ λ + 2 D2, (.9) where in addition to the standard term 4 F µνf µν we have also the term i 4 F µν F µν. The first term recalls the Yang-Mills euclidean action S YM [A] = 4g 2 d 4 x F µν F µν. The configurations that determine a finite value for this action can be caracterized on the basis of the omotopy class they belong to, labelled by the integer number (called winding number or Pontryagin index) k[a] = 32π 2 d 4 x F µνa F µνa, where F µν = 2 εµνρσ F ρσ. In each omotopy class, the Yang-Mills euclidean action reaches its minimum for a dual or antidual tensor, i. e. for gauge fields A µ such that F µν = F µν, as can be shown with simple inequalities which bring to S YM [A] 8π2 g 2 k[a]. For the class k[a] = 0, the Yang-Mills action has the trivial minimum S = 0 for A = 0, while for k > 0 the minimum is given by the instantons, which are non-trivial solutions of the equations of motion with finite action. For each winding number k, there is a multiplicity of vacuum states k. The transition amplitude between two different vacua states n and m, in the Feynman Path Integral approach, can then be calculated through the sum of the contributions of all the gauge fields belonging to the homotopy class ν, with ν = n m. In other words, the fact that the instantons are the minima of the Euclidean action allows to show that the semi-classical tunnelling amplitude between two non-homotopic vacuum state n and m can be computed as the sum over all the possible states (i. e. solutions for the equation of motion for the euclidean action) characterized by a winding number ν, each of them weighted by the exponential e SE/ (the equivalent of e is/ after the substitution t τ = it). This explains why only the solutions with finite action (just the instantons) contribute to the tunnelling amplitude. Then the true vacuum state can only be a superposition of different vacua state. Under a gauge transformation T characterized by a winding number of, the vacuum n gets transformed into T n = n + ; moreover, the gauge invariance implies that T commutes with the Hamiltonian, [T, H] = 0. This situation is strictly analogous to the quantum-mechanical problem of periodic potential: there, T is the translation operator, and the simultaneous eigenvalues of T and H are the Bloch waves. In analogy with that case, we then construct the true vacuum, called the θ-vacuum, as θ = n e inθ n, T θ = e iθ θ. With this condition, the system can be shown to be described by an effective lagrangian L eff = L + θ 6π 2 tr F µν F µν. We can define a vacuum as a Lorentz invariant stable configuration: this means that all space derivatives, and fields that are not scalars, in the vacuum configuration must vanish, then only the scalar field can have non-zero vacuum expectation value (usually denoted by vev).

12 8. Main features of Seiberg Witten theory After this short introduction to instantons, we return to the expression of the action.9. In order to let appear the F 2 term multiplied by the θ parameter (not to be confused with the θ α anticommuting variable), and with a real coefficient, and the instantonic term with the correct coefficient, we can introduce in the first integral the complex coupling constant τ = θ 2π + 4πi g 2, (.0) In this way we recover the real action [ g 2 d 4 x tr ] 4 F µνf µν iλσ µ µ λ + 2 D2 + θ 32π 2 d 4 x F µν F µν, (.) with the conventional renormalization on the F 2 term and the instanton number. The two actions.8 and. can be coupled in a unique action, which represents the N = 2 (two supersymmetric conserved charges) supersymmetric action. We only observe that the action containing the auxiliary fields can be simplified, using the equation of motion, as S aux = d 4 x 2 tr ([φ, φ]) 2, from which we can see that the bosonic potential is V (φ) = 2 tr([φ, φ]) 2 0. A ground state field configuration with V (φ 0 ) > 0 breaks supersymmetry. On the other hand, at the ground state, we require φ 0 0 in order the break the gauge symmetry group SU(2) and obtain massive fields through the Higgs mechanism. In this particular case, with the expression of the potential just introduced, we can see that we can fulfill both requirements if φ 0 0, but commutes with his adjoint φ 0, in order to have [φ 0, φ 0 ] = 0 and V (φ 0) = 0. The N = 2 supersymmetry can be rendered manifest with the introduction of a scalar superfield Ψ, involving the superfields Φ and W α introduced before (also a new set of anticommuting spinor variables θ is needed). With this substitution, one finds an action which depends only on Ψ, and not on Ψ. More generally, one can show that N = 2 supersymmetry constrains the form of the action to be 6π Im d 4 x d 2 θ d 2 θ F(Ψ), (.2) where the function F, called prepotential, depends only on Ψ: for this reason, it is called holomorphic. In order to resume the notation for the fields introduced in this section, we can rewrite the supersymmetric multiplets in the following way [3]:.2.2 Wilsonian effective action spin A i µ λ i α ψ βi 2 Wα i φ i 0 Φ i The Wilsonian effective action of the theory consists of an effective lagrangian description of the theory; it is obtained through the integration, with the tecnique of the Feynman Path Integral, only down to an infrared cut-off value in the momentum spectrum. The main physical effect of this approach is that we are left with a theory including only massless fields. In the adjoint representation, φ = 2 j φj σ j = j (aj + ib j )σ j, where a j and b j are real fields of the variable x. With a SU(2) transformation, we can always reduce to a (x) = a 2 (x) = 0. Then, when we are

13 .2. Supersymmetry in quantum field theories 9 looking for the ground state configuration, the vanishing of the commutator [φ, φ ] and the commutation properties of the self-adjoint Pauli matrices [σ i, σ j ] = i ε ijk σ k imply that b (x) = b 2 (x) = 0. Then, if we denote a = a 3 + ib 3, in the ground state configuration we have φ = 2 aσ3, and in the vacuum a must be a constant. Anyway, since a rotation around one of the SU(2) axis can still bring a to a, we also consider the gauge invariant quantity 2 a2 = tr φ 2. In the following, we will use the following definition for u and a, to include the cases in which quantum fluctuations around the ground state become relevant: u = tr φ 2, φ = 2 aσ 3. (.3) The complex parameter u distinguishes between gauge inequivalent vacua (i. e. ground state vacuum expectation values for the potential V (φ)). The manifold of definition for u, represented by the compactified complex plane with the exception of some singularities, is called the moduli space M of the theory. If the expectation value for the field φ is non vanishing in the vacuum, then, because of the Higgs mechanism, the SU(2) gauge symmetry is broken and two of the gauge fields become massive. In its essential lines, we can describe the mechanism as follows. The kinetic term for φ in the action, µ φ 2, includes (since µ is the covariant derivative) also the field A µ : then the expectation value for φ includes also a term proportional to A 2 that, after a suitable gauge transformation, allows to write a term in the lagrangian including a mass term for the vector field A i µ, i =, 2, with mass m = 2a. Similarly, due to the supersymmetric invariance, also the spinor fields ψ b, λ b, b =, 2, become massive with the same mass of A µ (since they all belong to the same supersymmetric multiplet, as seen in section.2). Then the fields A 3 µ, ψ 3, λ 3 and φ 3 remain massless, with a gauge symmetry broken from SU(2) to U(), but still with the N = 2 supersymmetry. In these conditions, the action for the theory reduces to 6π Im [ d 4 x d 2 θ F (Φ)W α W α + d 2 θ d 2 θ ] Φ F (Φ). (.4) One can see that, expanding the effective action.4, the term Im F (φ) appears to play the role of a metric on the moduli space. By comparison with the kinetic φ term, in which we introduced the complex coupling constant τ, we can set the second derivative of F(φ) to be equal to the complex coupling constant: Im F (φ) = Im τ = τ = F (φ). Then, in order to ensure the positivity of the metric, we must require that Im τ > 0: let us note that since F is not single-valued function, but a polymorphic function (we can think to F(u) as a section of a bundle over the moduli space), then we cannot apply the Liouville theorem and conclude that F is a constant over the compactified complex plane. We conclude this section by giving the expression for the prepotential F and its second derivative τ resulting from the combined tree-level and one-loop calculation for the region u : F(a) i 2π a2 ln a2 Λ 2, τ(a) i ) (ln a2 π Λ 2 + 3, (u ) (.5) where Λ 2 is a combination of constants and numerical factors chosen in order to fix the normalisation of F, called the dynamically generated scale. The presence of the logarithm in this expression clearly shows that the function F(a) has non-trivial monodromies for loops (a e 2πi a) around the singularity a = 0: then, as mentioned before, we could more formally think to F as a section of a bundle over the moduli space. Since the latter has some missing points (we have just introduced the singularity u = ), loops on the basis u-space are not trivially contractible over the fibration, then with loops on the complex u-plane we induce transformations of the section F.

14 0. Main features of Seiberg Witten theory.3 Duality in Seiberg Witten theory The formulæ.5 are valid only in the region of weak coupling (u ). Then we want to for an alternative description suitable to express the theory in a perturbative way: this can be managed exploiting a particular duality transformation. Let us consider the effective action after the integration over massive fields, eq..4, and the following transformation of the fields, Φ D = F (Φ), F D(Φ (.6) D ) = Φ, where Φ D is a dual field of Φ, and F D is a function dual to F. The corresponding dual value for a is a D = F a, since φ D = 2 a Dσ 3 and Φ φ =. These transformations can be seen as a Legendre transformation following from an hypothetical Hamiltonian F D (Φ D ) = F(Φ) ΦΦ D : in this interpretation, F would seem to represent the Lagrangian function of the coordinate Φ, while in Φ D = F Φ we could recognize the (complex) conjugate momentum: then the duality transformation.6 would correspond to the classical canonical transformation that exchanges the position and momentum of the particle, Q = p, P = q. In this case, the canonicity of the transformation ensures the conservation of the phase space measure, that in quantum field theory is represented by the measure over the possible paths DΦ. With the duality transformation.6, the second term of the action.4 remains invariant, while in the first term the complex coupling constant F is replaced by F (a) : in other words, τ τ. This transformation means that with the dual form of the fields we can pass from a strong coupling description, with an high value of the coupling constant that forbids the perturbative calculation, to a weak coupling one, since the new coupling constant is the inverse of the old one. This is a crucial point of Seiberg Witten theory, since it has allowed to successfully determine an effective action with a good perturbative approximation over each region of the whole moduli space: in fact, it is now sufficient to determine the values a, a D as functions of u in order to solve the N = 2 supersymmetric theory. This is what Seiberg and Witten achieved in their original paper [], that we will review in section 3.. We can recover a nice interpretation of the duality transformation.6 as a generalization of the wellknown electromagnetic duality for the classical electrodynamics. In fact, as we will show in section.4, the duality exchanges the electric and magnetic charges of the dyons forecast by the theory. Historically, the first to introduce the electromagnetic duality, noting a possible symmetric version of Maxwell equations, was Dirac. In fact, defining the magnetic 4-current j m µ = (φ m, j m ), and denoting by j e µ the electric 4-current, we can write the Maxwell equations in the form µ F µν = j ν e, ε µνρσ ν F ρσ = ν F µν = j µ m, with the introduction of the dual electromagnetic tensor F µν = 2 εµνρσ F ρσ. The previous equations are clearly invariant for the transformation F µν F µν, j µ e j µ m, usually called electromagnetic duality transformations, that can be written more evidently in terms of the fields and the charges as E B, e g, B E, g e, α = 4π e 2 4 α = e2 π ; the last transformation has been obtained exploiting the Dirac canonical quantization condition for the electric and magnetic fundamental charges eg = 2π c n, n Z. Then, the duality transformation clearly appears the analogous to the classical electromagnetic duality, since it exchanges magnetic and electric charges and brings the coupling constant for electrodynamics α to its inverse. We can conclude that the dual function a D (u) is then suitable to describe the theory in the region of strong coupling, in the patch near u = 0, where the effective action becomes the dual prepotential F D (a D ), in order to ensure the convergence of its perturbative expansion in terms of a D.

15 .3. Duality in Seiberg Witten theory We can rewrite the action.4 in the following way, in order to find the whole set of duality transformations: 6π Im d 4 x d 2 θ dφ D dφ W α W α + d 4 x d 2 θ d 2 θ (Φ Φ D Φ D 32πi Φ). We can rewrite the duality transformation.6 as ( ) ( ) ( ) ΦD 0 ΦD Φ 0 Φ ; (.7) from the previous form of the action, anyway, we can see that also the transformation ( ) ( ) ( ) ΦD b ΦD, b Z, (.8) Φ 0 Φ leaves invariant the action, since the second term does not change, while the first term can be seen to get only shifted by 2πbν, where ν is the integer istanton number, and a similar shift has not physical meaning because the action only appears as e is in the Feynman Path Integral. The two symmetries.7 and.8, with the ordinary matrix product, generate, as b varies over Z, the group SL(2, Z) of the 2 2 matrices with integers coefficients. This is then the complete group of duality transformations for Seiberg Witten theory. Duality represents a powerful tool, that moreover allows to prove in an alternative way the fact that the moduli space has exactly three singularities (which is one of the aims of chapter 3), as shown by Matone in [0]. We will show this proof in section at page U() R symmetry Another important symmetry of supersymmetric theories is the so called U() R symmetry: φ e 2iα φ θ e iα θ d 2 θ e 2iα d 2 θ W e iα W θ e iα θ d 2 θ e 2iα d 2 θ, that leaves invariant the action.2 only if F transforms into e 4iα F. Anyway, due to the instanton contributions, the corrections to the equation.5 for the prepotential bring to the following formula: F(a) i 2π a2 ln a2 Λ 2 + ( 2 4k a F k, (.9) Λ) where the terms F k are the instanton coefficients, that will be determined in section 3.2 following [6]. One can see that, under the U() R symmetry, the first term in.9 gets multiplied by e 4iα and adds a contribution of 8αν to the action, while the instantonic sum gets multiplied by e 4iα if (e 4iα ) 2k = : summing up, F e 4iα F if and only if α = 2πn 8, with n Z. This means that the instantonic and one loop contributions break the continuous U() R symmetry to a discrete Z 8 symmetry. This Z 8 symmetry anyway ulteriorly breaks if u = trφ 2 = 0, as happens in the case of breaking of SU(2) symmetry. In fact, if φ e iαπn 2, then u ( ) n u, and hence for u 0 the symmetry is broken to Z 4. We can fix this problem if we further impose the symmetry u u on the moduli space, in order to recover the whole Z 8 symmetry. k=0

16 2. Main features of Seiberg Witten theory.4 BPS states We include in this section, continuing the logical path introduced in section.3, a simplified discussion about the Bogomoln yi limit and BPS states, mainly following the exposition of Alvarez-Gaumé [, 2]. We begin by the formula that expresses a bound for the mass of a dyon (a both electrically and magnetically charged particle) in terms of its electric and magnetic charges q and g: M a g 2 + q 2, (.20) where a is the vacuum expectation value for the field φ. This formula essentially follows from the fact that in the rest frame the mass of a particle is given by the first entry T 00 of the energy-momentum tensor, that is equal to 2 (E2 + B 2 ) for electromagnetic lagrangian; using the Maxwell equations for the divergences of E and B we obtain after integration the charges over the whole tridimensional space. From this equation, with some manipulation, the inequality.20, called the Bogomoln yi limit, is obtained. This limit is saturated when V (φ) = 0: in this conditions, known as Bogomoln yi-prasad-sommerfeld (BPS) limit of the theory, the state (called BPS state) satisfies the condition.20 with the equality. The BPS bound appears very naturally in theories with N = 2 supersymmetry: in fact, in that case, the following inequality holds, M 2 2 Z 2, (.2) where Z is the central charge of the N = 2 SUSY algebra. For massless supermultiplets, the central charge must clearly be zero. We can essentially deline the meaning of central charge as follows. From the algebraic point of view, a supersymmetric theory arises as an extension of the algebra of the symmetries of a field theory, leaving invariant the algebra of the operators that commute with the hamiltonian of the system, e. g. the algebra of the generators of the Poincaré group in our case. Such operators cannot only be defined in terms of commutation relation, but also with anticommutation relations: these conditions determine the supersymmetry algebra. The result is that in the case of unextended SUSY (N = ) the only generator of the supersymmetry transformation (also called supersymmetry charge) anticommutes with itself, while in extended N = 2 SUSY we get two supersymmetric charges, and the anticommutation relations define the operator Z, called central charge. In this framework, we note that the fact that SUSY charges commute with the generators of the Poincaré group, including P 2, implies that all the fields entering a supermultiplet have the same mass. Returning back to formula.2,it is possible to show that at the level of the effective action the central charge can be written in the form Z = (an e + a D n m ). (.22) From an intuitive point of view, we can justify this formula as follows. The Dirac canonical quantization implies that the charges of a dyon can be expressed as n e q and n m g, where n e, n m are integers, q is the electron charge and g = 2π c e is the fundamental magnetic charge. The central charge, for a purely electrically charged state, takes the form Z = an e ; then duality implies that the analogous equation holds for a magnetic monopole, Z = a D n m. Since the central charge is additive, we recover formula.22. As seen in section.3, the full duality group is SL(2, Z), that acts over the column vector (a D, a) T with (a D, a ) T = M(a D, a) T. The conservation of the central charge then implies that the electric and magnetic charges are modified by the inverse of the matrix M: Z = ( ) ( ) a n m n D e = ( ( ) ) n a m n ad e M = ( ( ) n a m n e) = nm n e M. ( ) 0 For the case M = introduced in formula.7, we see that the duality transformation exchanges 0 the magnetic and electric charges of a dyon; in this sense, we can justify the previous statement that the duality transformation is a generalization of the electromagnetic duality. We conclude this section with a remark about the BPS spectrum of Seiberg Witten theory. For any singularity of the moduli space, there is a vacuum state corresponding to a dyon identified by the monodromy matrix associated to the vector (a D, a) T for a loop around the singularity, on the basis of the formula ( ) 2nm n M = e 2n 2 e 2n 2. m + 2n m n e

17 .4. BPS states 3 From the expression of the monodromy matrix that will be deduced in chapter 3, it follows that to any arbitrary non homotopically trivial loop around the singularities u 0, u 0, corresponds a dyon becoming massless. Moreover, it is possible to show that the only stable BPS states are the ones with (n m, n e ) relatively prime. We conclude this chapter with a quick review of the basic facts and notations that will be needed in chapter 3. The Wilsonian effective action for the N = 2 SYM SU(2) gauge theory can be determined in terms of the prepotential F. The Higgs mechanism, and the integration over massive fields, bring to the vacuum expectation values, for the massless field φ, defined as φ = 2 aσ3, trφ 2 = u: the latter, being the vacuum expectation value for the potential energy of the lagrangian for the auxiliary fields, is the coordinate for the moduli space M, which consists of the compactified complex plane with the exception of singularities to be determined. Suitable descriptions for the various patches of M are available choosing a(u) near u = and the dual coordinate a D (u) = F(a) a in the region of strong coupling. Then, in order to explicitely solve the theory, we only need to determine the form of the functions a(u), a D (u) on the basis of their non-trivial transformations for loops around the singularities of M. Moreover, we will be able to determine the instanton coefficients F k without the whole instanton calulations, but with the help of the framework of uniformization theory, that will be introduced in chapter 2.

18

19 CHAPTER 2 Riemann surfaces and uniformization theory After a quick introduction to the N = 2 SUSY gauge theory, we introduce basic facts inheriting complex analysis and Riemann surfaces that we will exploit in chapter 3 for the solution of the model. We begin in section 2. with the definition of Riemann surfaces and their principal properties, mainly from a topological point of view: this type of analysis leads to the powerful tool of uniformization theory (section 2.2) to study Riemann surfaces. This will be the framework for the solution of the model proposed by Matone, exposed in section 3.2 at page 35. Then in section 2.3 we will shortly introduce the basic facts about elliptic curves, the fundamental tool of the solution proposed by Seiberg and Witten in their original paper (section 3. at page 3). 2. General and topological aspects of Riemann surfaces We begin by the definition of Riemann surface. Definition 2. (Riemann surface). A Riemann surface is a one-complex-dimensional connected complex analytic manifold, i. e. a two-real-dimensional connected manifold Σ with a maximal atlas {U α, z α } such that the transition functions f αβ = z α z β : z β (U α U β ) z α (U α U β ) are holomorphic whenever U α U β. Such an atlas is called a set of analytic coordinate charts. As basilar examples of Riemann surfaces we can consider the following: Ĉ = C { }, the Riemann sphere, is the simplest compact Riemann surface. A set of analytic coordinate charts are z over U = C, and z over C \ {0} { }, with transition functions f 2 = f 2 = z. C, the complex plane. H = {z C Im z > 0}, the upper half plane. In complete analogy with the case of differentiable manifolds, we can define also for Riemann surfaces the analogue of differentiable applications and diffeomorphisms. Definition 2.2. A continuous mapping f : Σ Ω between Riemann surfaces is called holomorphic or analytic if, for every local coordinate (U, z) on Σ and (V, ζ) on Ω with U f (V ), the mapping ζ f z : z(u f (V )) ζ(v ) 5

20 6 2. Riemann surfaces and uniformization theory is holomorphic; if f is also one-to-one and onto, then f, f are called conformal. We define holomorphic function over Σ a holomorphic mapping of Σ into C; the ring of holomorphic functions will be denoted by H(Σ). We will then call meromorphic functions the holomorphic mapping into Ĉ, which constitute the field K(Σ) (since the inverse of a meromorphic function is still meromorphic). The notion of conformal application is very important in the theory of Riemann surfaces, since we are often interested in the description of surfaces up to conformal applications (in analogy with the concept of differential geometry of equivalence of differential structures induced by different atlases). As an example, we note that every atlas on Ĉ induces transition charts conformally equivalent to the ones previously described, f 2 (z) = f 2 (z) = z. 2.. Covering manifolds Almost the totality of Riemann surfaces are topologically non-trivial, and the analysis of their complex structure turns out to be the deeper tool to identify a surface. We recall the definition of universal covering manifold. Definition 2.3 (Universal covering manifold). A universal covering manifold Σ of Σ is a manifold such that:. there is a surjective local homeomorphism π : Σ Σ; 2. Σ is simply connected (the fundamental group of Σ is trivial); 3. every closed curve which is not homotopically trivial on Σ lifts to an open curve on Σ, which is uniquely determined by the curve on Σ and the initial point. When applying this notion to Riemann surfaces, we must preliminarly identify the simply connected ones. One can show that there are exactly three conformally distinct simply connected Riemann surfaces, which are just the three examples previously mentioned: the Riemann sphere Ĉ, the complex plane C and the upper half plane H. Furthermore, one could prove also that for any covering manifold Σ of Σ, the fundamental group π (Σ ) is isomorphic to a subgroup N of π (Σ). Moreover, there is a group G = π (Σ)/N of fixed point free 2 automorphisms of Σ such that Σ /G = Σ. In the case of universal covering manifold, from the second requirement of definition 2.3 it follows that N is the trivial group and G = π (Σ). In other words the map π of the definition 2.3 becomes a holomorphic mapping between Riemann surfaces and G becomes a group of holomorphic self mappings of Σ such that Σ/G = Σ, where Σ is one among Ĉ, C or H. We will call Σ/G a fundamental domain. The most important consequence of the previous theorems is that we can therefore study Riemann surfaces through the study of fixed point free discontinous groups of holomorphic self mappings of Ĉ, C and H. Let us examine in more detail the various possibilities. Ĉ C One can show that every non trivial self map of Ĉ has at least one fixed point, hence the sphere covers only itself. The desired maps are of the form z z + b, b C. One can show that a discontinuous subgroup of this group is either trivial or cyclic on one generator, or an abelian group with two generators. In the first case, the map is conformally equivalent to z z +, which identifies the cylinder (which is homotopically equivalent to the twice punctured sphere). In the second case, the group consists, without loss of generality, of mappings of the form z z + n + mτ, Im τ > 0, n, m Z. In this case the fundamental domain C/G identifies the parallelogram of vertices 0,, + τ, τ (with Im τ > 0) in which we identify the opposite edges (see figure 2.), thus the Riemann surface in question is the torus. From the definition of universal covering manifold, it follows that any Clearly N must be in this case a normal subgroup, otherwise we cannot define the quotient group. 2 As we will see in section 2.2, in the case of branching points the corresponding points on the covering manifold Σ are fixed points under the action of the group. These points are called orbifolds, and in the common use the surface Σ is nevertheless called a Riemann surface.

21 2.. General and topological aspects of Riemann surfaces 7 non homotopically trivial curve over the torus lifts to an open curve which links two points in C (belonging to different fundamental domains) identified by the group z z + n + mτ. Σ Σ 2τ Q + 2τ τ α P 0 + τ π Figure 2.: Universal covering manifold for the torus, and lifting of a non homotopically trivial closed curve α to an open curve α, with initial and final point P, Q identified by the group G. β P α H The most interesting Riemann surfaces have the upper half plane as universal covering manifold. One can show that all the holomorphic automorphisms of H can be represented as a particular cases of Möbius transformations z az + b cz + d, ( ) a b SL(2, R), (2.) c d where SL(2, R) represents the group of special (determinant equal to ) matrices 2 2 with real coefficients (in a generic Möbius transformation, the matrix would belong to SL(2, C)). This case will be further investigated in section Normal forms of Riemann surfaces In this section we will show how we can simplify the topological model of a Riemann surface through successive simplifications and edges identifications of a triangulation on the manifold (which is finite if the manifold is compact). This operation brings to the normal form of the surface, a polygon with appropriate identification of the edges which is topologically equivalent to the surface. Preliminarly, we review the concepts of homology and cohomology group. Let us consider a generic manifold Σ. We call zero-simplex any vertex (point) on the surface, one-simplex a generic edge (curve) linking two vertices, and two-simplex a triangle bounded by three edges. Hence a triangulation on the surface consists of a covering of the manifold through three-simplices on the surface. The orientation on the manifold induces an orientation on the triangles which can be used to orient the edges bounding the triangle: for example, the three simplex P, P 2, P 3 is bounded by the one-simplices P, P 2, P 2, P 3 and P, P 3. Let us denote with c n the additive group of n-chains, the linear combination (with integer coefficients) of n-simplices. Then we can define the operator δ : c n c n, which associates to an n-simplex its boundary, in the following way: δ P = 0, δ P, P 2 = P 2 P, δ P, P 2, P 3 = P, P 2 + P 2, P 3 P, P 3. Note that from the previous equations is evident that δ 2 = 0. If we denote by Z n the kernel of δ : c n c n, and by B n the image of c n+ in c n under δ, we can define the n-th simplicial homology group as H n (Σ) = Z n /B n.

22 8 2. Riemann surfaces and uniformization theory Thus with H (Σ) we denote the set of closed loops modulo the one-chains which constitute the boundary of a two-chain, i. e. the non homotopically trivial curves on the surface. The same analysis can be done for the dual set of the simplices, the set of the differential forms over the manifold. Given a generic manifold Σ of dimension n, we consider the space Λ k TP Σ of completely antisymmetric covariant tensors of rank k at the point P Σ; then Λ k TP Σ is a subset of k T P Σ. We define a differential k-form as a totally antisymmetric covariant tensor field; if we choose as a basis for Λ k the exterior product of the canonical -forms, dx i dx i k = signσ dx iσ() dx iσ(k), σ S k where S k is the group of permutations of k elements, then we can write a generic k-form as ω = ω i i k dx i dx ik. i < <i k We denote with Ω k (Σ) the space of differential k-forms over Σ. Then, in analogy with the previous definition of the operator δ, we define the exterior differential n d : Ω k (Σ) Ω k+ ω (Σ), dω = (x) i ik x i dx i dx j dx j k. i= j < <j k One can show that also in this case the exterior differential is involutary, d 2 = 0. Then we define closed form a differential k-form ω with dω = 0, and exact form a k-form ω if there exists a (k )-form α such that ω = dα. We observe that since d 2 = 0, every exact form is closed, while the converse holds only locally, without further hypothesis about the topological structure of Σ. These notions allow us to define an equivalence relation over Ω k (Σ), considering equivalent two closed differential forms if their difference is an exact form. The linear space of the classes of equivalence of closed k-forms on Σ is called the k-th De Rham cohomology space H k (Σ). The two concepts of homology group and cohomology group are clearly linked by a duality relation: in fact a n-form can be integrated over a n-chain. Hence a -form can be integrated over a finite union of smooth paths, while a 2-form ω can be integrated over a domain D Σ : ω = ω xy dx dy. D D Now we return back to the case of a compact Riemann surface. Given a triangulation on the surface, we can map each two-simplex to an euclidean triangle, and then proceed to simplify the figure removing admissible edges until we reach a simple polygon, which will have an even number of edges since each edge is identified with another one. A bit more precisely, to select the removable edges we must traverse all the edges (labelled with a, b,... ) of the triangulation writing in order the corresponding letter, adding an apex if the edge is passed through in a verse opposite to its orientation one. In this way we obtain the symbol of the polygon, which allows us to remove an edge a if the sequence aa appears in the symbol. At the end of this operation, we obtain the symbol a b a b a g b g a g b g : this is the normal form of the surface, and g is called genus of the surface (see figure 2.2). The genus of the surface is a topological invariant, which allows us to identify from a topological point of view the surface in question with the sphere with g handles. The polygon finally obtained can be seen as the surface itself, cut along the generators of the first simplicial homology group. From this poligon we can recover the original surface by pasting the corresponding edges in the verse suggested by the arrows. Furthermore, the polygon in the normal form can be considered as sitting in the universal covering manifold Σ of Σ: then the sides of the polygon represent the lifts to Σ of closed curves on the surface. In other words, each side of the poligon can be identified with the generators of the fundamental group π (Σ).